Group in which every endomorphism is trivial or an automorphism

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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The version of this for finite groups is at: finite group in which every endomorphism is trivial or an automorphism

Definition

A group in which every endomorphism is trivial or an automorphism is a (typically, nontrivial) group for which every endomorphism is either the trivial map (sending all group elements to the identity element) or is an automorphism.

Whether the trivial group is included or not is a matter of convention. We sometimes exclude it when comparing with other simple group-type properties.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite simple group finite, nontrivial, and no proper nontrivial normal subgroup finite simple implies every endomorphism is trivial or an automorphism |
finite quasisimple group finite, perfect, and inner automorphism group is simple finite quasisimple implies every endomorphism is trivial or an automorphism |
simple co-Hopfian group simple and not isomorphic to any proper subgroup simple co-Hopfian implies every endomorphism is trivial or an automorphism |

Weaker properties conditional to nontriviality

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
directly indecomposable group nontrivial and not expressible as a direct product of nontrivial groups Splitting-simple group|FULL LIST, MORE INFO
splitting-simple group nontrivial and has no proper nontrivial complemented normal subgroup every endomorphism is trivial or an automorphism implies splitting-simple |

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group in which every endomorphism is trivial or injective every endomorphism is trivial or injective; equivalently, it has no proper nontrivial endomorphism kernel |
Hopfian group every surjective endomorphism is an automorphism Group in which every endomorphism is trivial or injective|FULL LIST, MORE INFO
co-Hopfian group every injective endomorphism is an automorphism |